Abstract
Let \(X^{\bullet }=(X, D_{X})\) be a pointed stable curve of topological type \((g_{X}, n_{X})\) over an algebraically closed field of characteristic \(p>0\). Under certain assumptions, we prove that, if \(X^{\bullet }\) is component-generic, then the first generalized Hasse–Witt invariant of every prime-to-p cyclic admissible covering of \(X^{\bullet }\) attains maximum. This result generalizes a result of S. Nakajima concerning the ordinariness of prime-to-p cyclic étale coverings of smooth projective generic curves to the case of (possibly ramified) admissible coverings of (possibly singular) pointed stable curves. Moreover, we prove that, if \(X^{\bullet }\) is an arbitrary pointed stable curve, then there exists a prime-to-p cyclic admissible covering of \(X^{\bullet }\) whose first generalized Hasse–Witt invariant attains maximum. This result generalizes a result of M. Raynaud concerning the new-ordinariness of prime-to-p cyclic étale coverings of smooth projective curves to the case of (possibly ramified) admissible coverings of (possibly singular) pointed stable curves. As applications, we obtain an anabelian formula for \((g_{X}, n_{X})\), and prove that the field structures associated to inertia subgroups of marked points can be reconstructed group-theoretically from open continuous homomorphisms of admissible fundamental groups. Those results generalize A. Tamagawa’s results concerning an anabelian formula for topological types and reconstructions of field structures associated to inertia subgroups of marked points of smooth pointed stable curves to the case of arbitrary pointed stable curves.
Similar content being viewed by others
References
Bouw, I.: The \(p\)-rank of ramified covers of curves. Compos. Math. 126, 295–322 (2001)
Crew, R.: Étale \(p\)-covers in characteristic \(p\). Compos. Math. 52, 31–45 (1984)
Fried, M.D., Jarden, M.: Field Arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 11, 3rd edn. Springer, Berlin (2008)
Harbater, D.: Fundamental groups of curves in characteristic \(p\). Proceedings of the ICM (Zürich, 1994), pp. 656–666. Birkhäuser, Basel (1995)
Knudsen, F.: The projectivity of the moduli space of stable curves, II: the stacks \(M_{g, n}\). Math. Scand. 52, 161–199 (1983)
Mochizuki, S.: Semi-graphs of anabelioids. Publ. Res. Inst. Math. Sci. 42, 221–322 (2006)
Nakajima, S.: On generalized Hasse–Witt invariants of an algebraic curve. Galois groups and their representations (Nagoya 1981) (Y. Ihara, ed.), Advanced Studies in Pure Mathematics, vol. 2, pp. 69-88. North-Holland Publishing Company, Amsterdam (1983)
Ozman, E., Pries, R.: Ordinary and almost ordinary Prym varieties. Asian J. Math. 23, 455–477 (2019)
Pop, F., Saïdi, M.: On the specialization homomorphism of fundamental groups of curves in positive characteristic. Galois groups and fundamental groups. Mathematical Sciences Research Institute Publications, vol. 41, pp. 107–118. Cambridge University Press, Cambridge (2003)
Raynaud, M.: Sections des fibrés vectoriels sur une courbe. Bull. Soc. Math. France 110, 103–125 (1982)
Raynaud, M.: Sur le groupe fondamental d’une courbe complète en caractéristique \(p>0\). Arithmetic fundamental groups and noncommutative algebra (Berkeley, CA, 1999). Proceedings of Symposia in Pure Mathematics, vol. 70, pp. 335–351. American Mathematical Society, Providence, RI (2002)
Serre, J-P.: Sur la topologie des variétés algébriques en caractéristique \(p\). Symposium Intenacional Topologia Algebraica, Mexico, pp. 24–53 (1958)
Serre, J-P.: Trees. Translated from the French by John Stillwell. Springer, Berlin (1980)
Subrao, D.: The \(p\)-rank of Artin-Schreier curves. Manuscr. Math. 16, 169–193 (1975)
Tamagawa, A.: On the fundamental groups of curves over algebraically closed fields of characteristic \(>0\). Int. Math. Res. Notices 1999, 853–873 (1999)
Tamagawa, A.: On the tame fundamental groups of curves over algebraically closed fields of characteristic \(>0\). Galois groups and fundamental groups. Mathematical Sciences Research Institute Publications, vol. 41, pp. 47–105. Cambridge University Press, Cambridge (2003)
Tamagawa, A.: Finiteness of isomorphism classes of curves in positive characteristic with prescribed fundamental groups. J. Algebraic Geom. 13, 675–724 (2004)
Vidal, I.: Contributions à la cohomologie étale des schémas et des log-schémas, Thèse, U. Paris-Sud (2001)
Yang, Y.: On the admissible fundamental groups of curves over algebraically closed fields of characteristic \(p>0\). Publ. Res. Inst. Math. Sci. 54, 649–678 (2018)
Yang, Y.: Group-theoretic characterizations of almost open immersions of curves. J. Algebra 530, 290–325 (2019)
Yang, Y.: Tame anabelian geometry and moduli spaces of curves over algebraically closed fields of characteristic \(p>0\), preprint. http://www.kurims.kyoto-u.ac.jp/~yuyang/
Yang, Y.: On topological and combinatorial structures of pointed stable curves over algebraically closed fields of positive characteristic. http://www.kurims.kyoto-u.ac.jp/~yuyang/
Yang, Y.: On the averages of generalized Hasse–Witt invariants of pointed stable curves in positive characteristic. Math. Z. 295, 1–45 (2020)
Yang, Y.: Raynaud-Tamagawa theta divisors and new-ordinariness of ramified coverings of curves. J. Algebra 587, 263–294 (2021)
Yang, Y.: Moduli spaces of fundamental groups of curves in positive characteristic I, preprint, arXiv:2010.01806. See also http://www.kurims.kyoto-u.ac.jp/~yuyang/ for the most recent version
Yang, Y.: Moduli spaces of fundamental groups of curves in positive characteristic II, in preparation
Zhang, B.: Revêtements étales abeliens de courbes génériques et ordinarité. Ann. Fac. Sci. Toulouse Math. (5) 6, 133–138 (1992)
Acknowledgements
The main results of the present paper were obtained in January, 2019. The author would like to thank the referees very much for carefully reading the manuscript and for giving me many comments which substantially helped improving the quality of the paper. This work was supported by JSPS KAKENHI Grant Number 20K14283, and by the Research Institute for Mathematical Sciences (RIMS), an International Joint Usage/Research Center located in Kyoto University.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Yang, Y. Maximum generalized Hasse–Witt invariants and their applications to anabelian geometry. Sel. Math. New Ser. 28, 5 (2022). https://doi.org/10.1007/s00029-021-00720-8
Accepted:
Published:
DOI: https://doi.org/10.1007/s00029-021-00720-8
Keywords
- Pointed stable curve
- Admissible covering
- Generalized Hasse–Witt invariant
- Raynaud-Tamagawa theta divisor
- Admissible fundamental group
- Anabelian geometry
- Positive characteristic